The Zeldovich's prediction of analpole, a property related to the nuclei that undergo
\(\beta \) -decays [1]}, eventually led Dubovik and
coworkers to the discovery of a whole new class of toroidal multipoles in
electrodynamics [2]}, [3]}.
Since then, the physical significance of the toroidal multipoles have been highlighted
in practically all areas of physics, from particle [4]}, [5]}, [6]}, [7]},
nuclear [8]}, [9]}, and
atomic physics [10]}, to condensed matter
systems [11]}, [12]}, [13]}, [14]}, [15]}, [16]}, [17]}, [18]}, [19]}
and metamaterials [20]}, [21]}, [22]}, [23]}, [24]}, [25]}.
The toroidal dipole is the lowest order term in this class and a quantum operator
\(\hat{{\bf T}}\)  of components \(\hat{T}_i\) , \(i = 1,2,3\) , has been associated with
it [10]}, [27]}, [28]}, [29]}.
When defined in the whole \(\mathbb {R}^3\)  space, the operators \(\hat{T}_i\)  are hypermaximal
(i.e., they have multiple self-adjoint extensions) [27]}.
In finite systems, the operators \(\hat{T}_i\)  may be split into components along local
curvilinear systems of coordinates.
In Ref. [29]} it was shown that, if some of these coordinates are cyclic, then the components along them are self-adjoint, corresponding, therefore, to quantum observables.
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